Proposition 5 in [1]} ([1]}) indicates that multivariate functional data have a finite Karhunen–Loève representation if and only if all univariate elements have a finite Karhunen–Loève representation:
\({{Y}}^{(j)}_i(t^{(j)}_i)={X_i^{(j)}}(t^{(j)}_i)+{\epsilon }_i^{(j)}={\mu _i^{(j)}}(t^{(j)}_i)+\sum _{l=1}^{\infty } \xi ^{(j)}_{i,l} \phi ^{(j)}_{i,l}(t^{(j)}_i)+{\epsilon }_i^{(j)}, ~~~i=1,\ldots ,n,~~~j=1,\ldots ,p,\) 
